Bridging the Gap to University Mathematics

2009·1.59 MB·english

by  Martin Gould

Bridging the Gap to University Mathematics
Martin Gould • Edward Hurst
Bridging the Gap
to University Mathematics
123
ISBN:978-1-84800-289-0 e-ISBN:978-1-84800-290-6
DOI:10.1007/978-1-84800-290-6
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Preface
Mathematicshasalwaysbeenanexcitingchallengeforbothofus.Evenbefore
university, we thoroughly enjoyed getting to grips with calculus and meeting
stuff like vectors and mechanics for the first time. The only problem that we
both found when we first started our degrees was that amongst our new peers,
different people had very different mathematical backgrounds. Some had done
“single maths,” some “double,” and some an entirely different syllabus alto-
gether. We quickly learned that a mathematics degree requires mastery of a
whole range of ideas: including those that you haven’t studied before.
The aim of this book is in no way to teach you everything that you would
learn in the first year of a mathematics degree. Instead, our aim was to write
a book that you could read before going to university that would give you a
solid foundation on which to build all of the new skills that you will acquire.
That way, when you actually arrive at university you will have much more
timeforalloftheotheramazingthingsthatbeingastudentoffers,ratherthan
havingtospendhourslookingupsomethingthatyoucouldeasilylearninafew
minutesfromastraightforwardbooklikethis.Tothisend,wehaveincludedan
appendix of loads of formulae and identities so that you can spend your nights
partying rather than searching in the library for the integral of tanx.
I suppose being young means being radical, so we’ve written this book
backwards. Not crazy, “read in the mirror” backwards, but the chapters are
set out in the reverse of what you are probably used to seeing. Each chapter
is designed to be a completely stand-alone entity, and chapters always start
with some questions. Our reason for doing this is so that if you see a chapter
title about something with which you are familiar, you can dive straight into
some questions then head off to the next chapter without having to read any
explanation of the topic. If you really want to fly through the material, the
vi Preface
first ten “Test Yourself” questions of each chapter are designed to cover the
key points. If you can score close to full marks on these, you’re doing pretty
well. If what the questions are asking looks foreign to you, work through the
chapterandtherestoftheexercisesandcomebacktotheselast.Allbeingwell,
you’ll be able to do them within a reasonably short time. If you have already
studiedalotof“pre-university”maths,youmaywellhaveagoodknowledgeof
quiteafewoftheearlierchapters’contents.Ifyouhaven’t,therewillobviously
be more chapters where you’ll be starting afresh. For those readers taking
the International Baccalaureate (or any other equivalent qualification) the key
to success is exactly the same – study what you need to, pass over what you
don’t.Inanycase,thelatterchaptersmovefurtherawayfromschoolandcollege
mathematicsandtowardsdegree-stylethinking,soasthebookprogressesthere
is certainly something for everyone.
Pleasedon’teverbedisheartenedifyou’refindingsomethingsdifficult.We
wrote this book in the hope that it could help prepare you as fully as possible
foryourstudies,sothatyoucanhavethebesttimeatuniversity,bothinterms
of academic achievement and your student life as a whole. There may be times
when you would like more practice with a new skill, and so we thoroughly
recommend that in these instances you search out some more questions to do,
either from the Internet or from other books.
On the whole, we hope that regardless of whether you are a student who
has just decided to study maths further, or you are someone just about to set
off to university, this book will serve you well as a single, cohesive guide that
draws on your knowledge thus far and helps shape it so that you are ready to
tackle the challenges of the mathematics ahead.
Martin Gould and Edward Hurst
Contents
Acknowledgements ............................................. xi
1. Inequalities ................................................. 1
1.1 What Are Inequalities?.................................... 2
1.2 Using Graphs ............................................ 4
1.3 Critical Values ........................................... 7
2. Trigonometry, Differentiation and Exponents ............... 13
2.1 Some Identities........................................... 14
2.2 Differentiating ........................................... 19
2.3 Exponents and Logarithms ................................ 23
3. Polar Coordinates .......................................... 31
3.1 A Different Slant ......................................... 32
3.2 Lines and Circles ......................................... 36
3.3 Moving on Up ........................................... 41
4. Complex Numbers.......................................... 49
4.1 Numbers ................................................ 50
4.2 Working with Complex Numbers ........................... 52
4.3 Tips and Tricks .......................................... 56
5. Vectors ..................................................... 61
5.1 Reinventing the Wheel .................................... 62
5.2 A Different Approach ..................................... 67
viii Contents
5.3 The Cross Product ....................................... 70
6. Matrices.................................................... 73
6.1 Enter the Matrix ......................................... 74
6.2 Multiplication and More................................... 78
6.3 Determinants and Inverses................................. 83
7. Matrices as Maps........................................... 89
7.1 Over and Over ........................................... 90
7.2 Old Friends.............................................. 95
7.3 Eigenvalues and Eigenvectors ..............................104
8. Separable Differential Equations ............................113
8.1 Repetition Is the Key to Success ...........................114
8.2 Separation of Variables....................................120
8.3 Combining the Tools......................................123
9. Integrating Factors .........................................127
9.1 Troubling Forms .........................................128
9.2 Productivity .............................................131
9.3 The Finishing Line .......................................133
10. Mechanics ..................................................137
10.1 Where You Want to Be ...................................138
10.2 Faster! Faster! ...........................................145
10.3 Resolving Forces .........................................149
11. Logic, Sets and Functions...................................157
11.1 Set Notation.............................................158
11.2 Logical Equivalence.......................................162
11.3 Functions................................................166
12. Proof Methods .............................................171
12.1 Proof by Induction .......................................172
12.2 The Principle of Induction.................................173
12.3 Contrapositive Statements.................................178
13. Probability .................................................183
13.1 Turn that Frown Upside-Down .............................184
13.2 Solving Probability Problems ..............................187
13.3 Conditioning.............................................191
Contents ix
14. Distributions ...............................................195
14.1 Binomial Events..........................................196
14.2 Poisson Events ...........................................198
14.3 Using Binomial and Poisson Models.........................201
15. Making Decisions...........................................207
15.1 A Whole New Probability .................................208
15.2 Story Time ..............................................212
15.3 Decision Problems........................................215
16. Geometry ..................................................221
16.1 Old Problems, New Tricks .................................223
16.2 Proof ...................................................226
16.3 3D Geometry ............................................235
17. Hyperbolic Trigonometry...................................241
17.1 Your New Best Friends....................................242
17.2 Identities and Derivatives..................................246
17.3 Integration ..............................................250
18. Motion and Curvature......................................253
18.1 Loose Ends or New Beginnings?............................254
18.2 Circular Motion ..........................................261
18.3 Curves ..................................................268
19. Sequences ..................................................273
19.1 (Re)Starting Afresh.......................................274
19.2 To Infinity (Not Beyond) ..................................276
19.3 Nothing at All ...........................................281
20. Series.......................................................285
20.1 Various Series............................................286
20.2 Harmonics and Infinities...................................289
20.3 Comparison Testing ......................................292
A. Appendix...................................................295
A.1 Inequalities ..............................................295
A.2 Trigonometry, Differentiation and Exponents.................296
A.3 Polar Coordinates ........................................296
A.4 Complex Numbers........................................297
A.5 Vectors..................................................297
A.6 Matrices ................................................298
A.7 Matrices as Maps.........................................298
x Contents
A.8 Separable Differential Equations............................299
A.9 Integrating Factors .......................................299
A.10Mechanics ...............................................300
A.11Logic, Sets and Functions .................................301
A.12Proof Methods ...........................................302
A.13Probability ..............................................303
A.14Distributions.............................................304
A.15Making Decisions.........................................304
A.16Geometry ...............................................304
A.17Hyperbolic Trigonometry..................................305
A.18Motion and Curvature ....................................305
A.19Sequences ...............................................305
A.20Series ...................................................306
A.21Pure: Miscellaneous.......................................306
A.22Applications: Miscellaneous................................307
B. Extension Questions ........................................309
C. Worked Solutions to Extension Questions...................313
Solutions to Exercises ..........................................323
Index...........................................................343
Acknowledgements
This project would never have been possible without the continued support
of our friends and families, to whom we wish to extend our sincerest thanks.
WewouldespeciallyliketothankHannahKimberleyforherfantastic,diligent
work in checking and rechecking everything we did; Professor Ian Stewart for
his expert knowledge in the field; Giulian Ciccantelli and Richard Revill for
theirmathematicalandartisticinsights;JosephCarter,MasoumehDashtiand
Hannah Mitchell for their input at the crucial collaboration phase; and our
parents Tina and James Gould and Helen and Tim Hurst for putting up with
a summer of us writing all the time – and eating all their food. We couldn’t
have done it without each and every one of you.